⚛️ quantum physics

Entropy Density Benchmarking of Near-Term Quantum Circuits

This paper introduces a novel benchmarking methodology based on entropy density accumulation to model noise in near-term quantum devices, enabling more accurate determination of circuit volume thresholds for quantum advantage than existing techniques.

Original authors: Marine Demarty, James Mills, Kenza Hammam, Raul Garcia-Patron

Published 2026-08-25

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Marine Demarty, James Mills, Kenza Hammam, Raul Garcia-Patron

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Modern computers are powerful, but they are not perfect. They make mistakes, and as tasks get harder, those mistakes pile up until the answer becomes useless. Quantum computers, the next generation of machines that use the strange rules of physics to solve problems, face this same issue but with far greater intensity. Because they are so sensitive to their environment, the information they hold tends to blur and fade away very quickly. This fading is measured by a concept called entropy, which essentially tracks how much order has turned into disorder. When a quantum computer works, it tries to keep its information pure and ordered. However, noise from the outside world constantly pushes the system toward a state of total confusion, where every possible answer is equally likely. The central challenge for scientists today is to figure out exactly how much work a quantum computer can do before this noise swamps the signal, rendering the machine no better than a random guess.

A team of researchers at the University of Edinburgh, working with collaborators in France and the United Kingdom, has developed a new way to measure this limit. Instead of trying to run a complex, real-world problem on a quantum computer to see if it works, they created a method to predict the point of failure by tracking how much disorder accumulates as the computer processes information. They call this "entropy density," a measure of how much confusion exists for each piece of the machine's memory. By building simple models of how this confusion grows and testing them against real experiments, the team found a way to calculate a hard ceiling on the size of any task a current quantum computer can handle. Their work suggests that for many practical problems, the limit is much lower than previously thought, meaning we may need to wait for better hardware before these machines can truly outperform the best classical computers.

The researchers focused on a specific type of quantum algorithm known as a variational quantum algorithm, which is a popular method for solving optimization problems on today's noisy machines. These algorithms work by layering simple operations on top of each other, like stacking blocks, to build up a solution. The team wanted to understand what happens to the information as these layers are added. They started by running simulations on classical computers, modeling a quantum circuit with a specific number of layers and qubits, the basic units of quantum memory. They introduced a standard type of error, known as depolarizing noise, which randomly scrambles the state of the qubits. As they added more layers to the circuit, they watched the entropy density rise. They found that the disorder grew steadily, and for larger systems, it converged toward a maximum level of confusion very quickly. This maximum level represents a state where the quantum computer has lost all useful information and is effectively just a bag of random noise.

To make sense of this rapid growth, the team developed a simple mathematical model based on the idea of global depolarizing noise. This model assumes that after the entire circuit finishes its work, the whole system is hit by a single, uniform wave of noise that mixes everything together. Surprisingly, this simple assumption, which ignores the complex details of how errors happen at every single step, matched the results of their detailed simulations very well. It provided a clear rule: as the circuit gets deeper and wider, the entropy density climbs until it hits a threshold where the quantum advantage disappears. The researchers then took this model to the real world to see if it held up. They used a superconducting quantum processor from Rigetti, a company that builds these machines, to run the same circuit layers they had simulated. They measured the purity of the output state, which is the inverse of entropy, using a technique called classical shadows. This method allowed them to estimate the disorder in the system without needing to measure every single detail, which would be impossible for larger machines.

The experimental results showed that the real machine behaved similarly to the model, but with a twist. The entropy in the actual device grew faster than the simple model predicted. The researchers investigated why this was happening and found that the standard model missed a crucial factor: the time it takes for the qubits to relax back to their resting state, known as T1 relaxation. When they added this physical reality into their model, the predictions aligned much better with the experimental data. The machine was losing information not just because of random noise, but also because the qubits were naturally decaying over time. This refinement was critical because it meant their model could accurately predict the performance of the hardware. They discovered that even with these refinements, the global depolarizing model remained a useful tool, acting as a conservative lower bound that guaranteed the machine would fail before reaching the predicted limit.

With a validated model in hand, the team applied their findings to a specific benchmark for quantum advantage: the MAX-CUT problem, a classic challenge in computer science where one tries to divide a network into two groups to maximize the connections between them. They combined their entropy model with existing knowledge about how well classical computers solve this problem. By calculating the point at which the quantum computer's entropy would become so high that its answer would be worse than the best classical solution, they established a new, stricter limit on the circuit size. For a typical modern quantum processor with a two-qubit gate error rate of one in a thousand, they found that the machine loses its advantage once the circuit reaches a depth of about 110 layers. This is a much lower threshold than previous estimates suggested, indicating that the window for quantum advantage is narrower than many had hoped.

The significance of this work lies in its ability to bridge the gap between the low-level physics of the machine and the high-level performance of the application. Previously, scientists had to choose between measuring the quality of individual gates or running a full application to see if it worked. This new method allows them to predict the outcome of an application simply by measuring how much entropy accumulates in the circuit. It provides a clear, quantitative way to say, "If you try to solve a problem this big, the noise will win." While the researchers acknowledge that their model is a simplification and that real devices have other types of errors, such as crosstalk between qubits, they argue that this approach offers a reliable and practical way to set expectations. It tells us that for current hardware, the path to solving complex real-world problems is blocked by noise much sooner than we might have guessed, and that future breakthroughs will depend on building machines that can keep their information pure for longer.

Technical Summary: Entropy Density Benchmarking of Near-Term Quantum Circuits

Problem Statement
Current and near-term quantum devices operate in the Noisy Intermediate-Scale Quantum (NISQ) regime, characterized by imperfect qubit control and environmental noise that prevents fault-tolerant computation. While these devices exceed the 50-qubit threshold where classical simulation becomes difficult, their practical utility is limited by error accumulation. A critical challenge is the isolation of benchmarking protocols: gate-level, circuit-level, and application-level benchmarks often operate in silos, lacking a unified framework to translate low-level noise characteristics into high-level predictions about quantum advantage. Specifically, there is a need for a method to determine the circuit volume threshold above which a quantum device can no longer outperform classical solvers for specific optimization problems.

Methodology
The authors propose a benchmarking methodology centered on entropy density accumulation as a performance metric. The approach bridges circuit-level noise modeling with application-level performance bounds through three main steps:

Heuristic Modeling: The authors develop simple heuristic models to describe how entropy accumulates in Variational Quantum Algorithm (VQA) circuits. They investigate a hardware-efficient VQA ansatz under local depolarizing noise. To simplify the analysis, they propose a global depolarizing (white-noise) model applied after the ideal circuit evolution. This model relates the global depolarizing probability $P$ to the circuit width ( $n$ ), depth ( $D$ ), and local noise parameters ( $p_1, p_2$ ) by equating the probability of no error occurring in both models.
Experimental Validation: The model is tested against experimental data from Rigetti's Aspen-M-3 superconducting quantum processing unit (QPU). The authors utilize the classical shadows protocol (specifically the Pauli measurement variant) to estimate the purity and second-order Rényi entropy density of the output state. They address systematic errors by comparing different measurement randomization strategies and incorporating calibration data for gate errors, readout errors, and $T_1$ relaxation into their simulations.
Application-Level Bounding: The validated entropy accumulation model is combined with a theoretical framework from prior work [23] to derive bounds on quantum advantage. By establishing a lower bound on the energy density solution of a quantum device as a function of its entropy density, the authors determine an entropy threshold ( $c$ ) beyond which the quantum device cannot outperform state-of-the-art classical solvers. This threshold is then mapped back to circuit size constraints (width and depth).

Key Results

Model Accuracy: Numerical simulations demonstrate that a global depolarizing model provides a good heuristic description of entropy accumulation for small systems and specific circuits, capturing the asymptotic convergence toward the maximally mixed state.
Experimental Discrepancies: Experimental data from the Aspen-M-3 QPU shows a faster entropy accumulation than predicted by a simple global depolarizing model based solely on gate calibration. While incorporating measurement errors and $T_1$ relaxation explains a significant portion of this gap, a residual difference remains, likely due to coherent errors, crosstalk, or complex damping not captured by the simple model.
Refined Thresholds: Despite the model's simplicity, the authors demonstrate that it provides a conservative lower bound on the available circuit volume. When applied to the MAX-CUT problem (specifically the G12 instance with 800 nodes), the derived depth threshold for losing quantum advantage is tighter than previous estimates. For a two-qubit gate error rate of $p_2 = 10^{-3}$ , the authors predict that circuits with a depth $D \geq 110$ are already in a regime of certified classical superiority.
Scaling Behavior: The analysis reveals that for large system sizes ( $n \to \infty$ ), the depth threshold for quantum advantage saturates at a constant value determined by the noise parameters and the classical solver's performance, specifically $D^* \approx \frac{c \ln(2)}{2p_2}$ .

Significance and Claims
The paper claims to provide a proof-of-principle demonstration of a novel benchmarking methodology that effectively bridges the gap between circuit-level and application-level protocols. The primary significance lies in the ability to:

Estimate the circuit size threshold above which quantum advantage is unattainable for a given problem without needing to run the full application on the quantum hardware.
Outperform existing techniques in bounding the circuit size threshold, offering more stringent and accurate predictions of when a device loses its quantum advantage.
Utilize entropy density as a central tool that complements existing benchmarking techniques, providing a unified view of how noise degrades computational utility.

The authors remain modest regarding the model's completeness, acknowledging that while global depolarizing noise is a useful first approximation, a more comprehensive model incorporating amplitude damping and coherent errors is necessary for tighter predictions. They also note that for larger systems, the classical shadows protocol may become impractical due to sampling costs, suggesting the SWAP test or other relaxation techniques as future alternatives.

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